Mode locking of spin waves excited by direct currents in microwave nano-oscillators

A spin-wave theory is presented which explains the frequency pulling and mode locking observed when two closely spaced spin-transfer nanometer-scale oscillators with slightly different frequencies are separately driven in the same magnetic thin film by spin-polarized carriers at high direct-current densities. The theory confirms recent experimental evidence that the origin of the phenomena lies in the nonlinear interaction between two overlapping spin waves excited in the magnetic nanostructure.